Large Time Behavior in a Multidimensional Chemotaxis-Haptotaxis Model with Slow Signal Diffusion

This paper studies the coupled chemotaxis-haptotatxis system $u_t= \Delta u - \chi \nabla \cdot (u\nabla v) - \xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), x\in \Omega, \, t>0; v_t=\Delta v-v+u, x\in \Omega, \, t>0; w_t=-vw, x\in \Omega, \, t>0$, in a smoothly bounded domain $\Omega\subset\mathbb{R}^n$, $n\le 3$, with zero-flux boundary conditions, where $\chi,\xi$, and $\mu$ are given positive parameters. It is shown that whenever the initial data $(u_0, v_0, w_0)$ are nonnegative and suitably regular fulfilling $u_0\not\equiv 0$ and $w_0\le 1$, the third solution component $w$ decays asymptotically in $L^\infty(\Omega)$. Moreover, under the fully explicit condition $\mu>\frac{\chi^2}{8}$ the solution $(u, v, w)$ exponentially stabilizes to the constant stationary solution $(1,1,0)$ in the norm of $L^\infty(\Omega)$ as $t\to \infty$.